Differential Form

First uploaded on 2022/09/22
Last Updated on 2024/05/07
Copyright(C)2022-2024 jos <jos@kaleidoscheme.com> All rights reserved.

The basic equation for Radiatively Driven Circulation (RDC), Eq. (9), \begin{eqnarray} \frac{\partial}{\partial x} \left(\rho {u}_{R}\right) +\frac{\partial}{\partial y} \left(\rho {v}_{R}\right) = - \frac{\partial}{\partial z} \left[ \rho {\left( \frac{g}{{c}_{p}} - \Gamma \right)}^{-1} {\left. \frac{\partial {T}}{\partial t} \right|}_{R} \right] \nonumber \end{eqnarray} is expressed in differential form. The right-hand side contains the vertical derivative of the expression including the radiative cooling rate -δT/δt|_R. Perhaps someone familiar with traditional meteorological dynamics might wonder if such a small value can really contribute to the transport around a cumulus cloud, or if they should keep focusing on the intense dynamical motion. It is because the magnitude of each term determines its effect in the time development equations that frequently appear in traditional meteorological dynamics.

The mechanism of RDC is completely different from the mechanism of traditional meteorological dynamics. Equation (9) only describes a local spatial relationship, containing no time derivative terms. Therefore, it is not solved as a time development problem, as is often the case in traditional meteorological dynamics. But Eq. (9), together with its counterpart equation (10) \begin{eqnarray} \frac{\partial }{\partial x} \left(\rho {v}_{R}\right) -\frac{\partial}{\partial y} \left(\rho {u}_{R}\right) = 0 \label{rot0} \nonumber \end{eqnarray} for example, is solved moment by moment as a boundary value problem under appropriate boundary conditions. That is, the horizontal flow field of RDC (ρu_R, ρv_R) is obtained by integrating these differential equations over a large x-y horizontal plane. It would be easy to imagine that, the value of the right-hand side of the basic equation for RDC (9) may be small, but if the space of integration is large enough, a flow large enough to account for the transport from cumulus clouds can be created.

Finally, it is important to emphasize that RDC is obtained by solving a boundary value problem, whereas most problems of traditional meteorological dynamics are solved using time development equations. This fact that the solution methods are fundamentally different means that RDC cannot be represented by any traditional meteorological dynamics such as dynamical detrainment.